Quantitatives 问题 大, 系数, 投资组合

shakaronnie

 已知一个投资组合P0的方差为0.024，如果有一个股票S相对投资组合的关联系数为0.038（<1），现在将S添加到P0中形成一个新的投资组合P1，对于P1一下结论正确么？

1）P1的非系统性风险小于P0

2）P1的方差不可能大于0.024

 

根据答案，因为S与P0关联系数小于1，所以P1的非系统性风险小于P0，P1的方差也不可能大于P0。

但是我不理解，如果S本身方差很大（比如10），且S在P1中占的比例又很大（比如99%），那么P1的方差肯定会很大吧（接近于10）。所以不理解。

 

以下是原题，贴出来方便参考。同时如果我理解题目有误，也欢迎大家纠正。谢谢！

 

 

Mital Tiene’s investment portfolio currently consists of stocks in two companies, 40 percent in Drysdahl Banking and the remaining amount in Clampett Oil. Performance measurement information for these two stocks is given in the table below:fficeffice" />

 

Stock                                   Expected Return              Standard Deviation

Drysdahl Banking                      10.50%                               8.5%

Clampett Oil                               16.55%                               25.0%

 

The covariance between the two stocks is 0.001. Tiene is considering adding a third stock, Hilbilee Investors. Hilbilee Investor’s correlation coefficient with the current portfolio is 0.38.

Which of the following statements is FALSE?

A) As Tiene diversifies, he will reduce the portfolio's unsystematic risk.

B) The standard deviation of returns for the current portfolio is 15.5%.

C) With Hilbilee added to the portfolio, the variance could be 0.026.

D) The expected return of Tiene's current portfolio is approximately 14.1%.

 

 

Reference Answer is C。

leekeewen

Hi Shakaronnie, you are actually right. I re-considered this question and did a few tests...the variance of the new portfolio could be any numbers since we don't have weights plus the standard deviation of that third stock....If I'm not right, I'll let u know asap... thanks!

shakaronnie

Zray,

 

Thank you for your answer. I believe your understanding of corelation is correct. And there must be something wrong with my understanding. But I can not figure it out. Can you help?

 

I'd describe my understanding starting from your conclusion as the following:

 

(1). according to your conclusion, if there are two stocks, whose Corr is less then 1, then portfoilio consisting of the two stocks should have a risk (standard deviation) less the both of the risk of the two single stock. (because each stock will diversify away some of the other's unsystematic risk) 

 

is my opinion correct?

 

(2). if (1) is correct, it may lead to a contradiction.  

 

let's just use the given data in the question as an example, and ignore the additional third stock. The first two stocks have a standard deviations of 8.5% and 25% respectively, and their correlation coeffienct is 0.001/(0.085*0.25) = 0.047, which is less than 1. So from (1), we will have that a portfolio of the two stocks should have a standard deviation less than min(8.5%, 25%), that is < 8.5%. However, from the given data, we know that the portfolio's standard deviation is 15.5%, which is between the standard deviations of the  two single stock.

 

Intuitively, it means first we have a portfolio consisting of only one 8.5% standard deviation stock. then we add another stock into the portfolio whose Corr with the former stock is 0.047 and whose standard deviation is 25%. But the newly added stock does not diversify away the unsystematic risk from the original single-stock portfpolio, instead, it raises the risk from 8.5% to 15.5%. This situation is contradictory to the conslusion offered by you as well as how the answer goes.

 

Could you kindly help looking for the incorrect places in the statement above. Thanks a lot.

 

zray

Since the correlation between that stock and current portfolio is 0.38, then the new portfolio that combines the current one and the new stock should have a std dev less than the current value, which is 15.5%. Let's square root 0.026 and get the new std dev=16.12%, which is larger than 15.5%. It's impossible cuz based on their correlation, adding that stock should be able to diversify away some of the unsystematic risk, no matter what weight and std dev of that stock to be added to the portfolio.

Thanks.